Basic English Institute

THE DEVELOPMENT OF MATHEMATICAL LOGIC
by
P. H. NIDDITCH

7 . The Algebra of Logic after Boole : Jevons, Peirce and Schroeder. . .44
7.1 . Later Developments of Boole's Algebra . . .44
7.2 . The Work of Jevons. . .44
7.3 . Caomparison of Boole's and Jevons's Systems. . .46
7.4 . Peirce's Points of Agreement with Boole and Jevons. . .48
7.5 . Peirce's Use of the 'A Part of ' Relation. . .49
7.6 . Peirce's Theory of Relatons. . .50
7.7 . Peirce's Logic of Statement Connections. . .52
7.8 . Peirce's Quantifier Logic. . .55
7.9 . Schroeder's Algebra of Logic. . .56
7.10. Whitehead's and Huntington's Work on the Algebra of Logic. . .58
8 . Frege's Logic. . .59
8.1 . The Mathematics of Logic and the Logic of Mathematics. . .59
8.2 . Reasons Why Frege's Writings were Undervalued. . .60
8.3 . Frege's View of the Relation between Logic and Arithmetic. . .61
8.4 . Frege on Functions. . .63
8.5 . Functions as Rules. . .63
9 . Cantor's Arithmetic fo Classes. . .66
9.1 . Finite and Enumeral Classes . . .66
9.2 . Cantor's Cardinal Arithmetic . . .68
9.3 . Cantor's Ordinal Arithmetic . . .69
9.4 . Burali-Forti's and Russell's Discoveries. . .69
10 . Peano's Logic. . .73
10.1 . Implication and Mathematics. . .73
10.2 . Peano's Purpose in his Logic. . .74
10.3 . Some of Peano's Discoveries and Inventions. . .75
10.4 . Pean's School of Metamathematics. . .76
11 . Whitehead and Russell's 'Principia Mathematica'. . .77
11.1 . Short account of the three parts of 'Principia Mathematica'.. . .77
11.2 . Relation between Russell's 'Principles' and 'Principia Mathematica'. . .78
11.3 . Russell's way of getting consistency in class theory.. . .78
12 . Mathematical Logic after "Principia Mathematica" : Hilbert's Metamathematica. . .79
12.1 . The Growth of Logic after 'Principia Mathematica'. . .79
12.2 . The Structure of an Axiom System. . .80
12.3 . Post's, Hibert and Ackermann's, and Gordel's Demonstrations for
        'Principle Mathematica' Systems
. . .81
12.4 . Brouwer's Doubts and Hilbert's Metamathematics. . .82
12.5 . Goedel's Theorem in the Metamathemaics of Arithmetic. . .84
12.6 .The Theory of Recursive Functions. . .84
 
Further Reading. . .86
Index. . .87
viii

7   THE ALGEBRA OF LOGIC AFTER BOOLE : JEVONS, PEIRCE AND SCHROEDER

7.1. Later developments of Boole's algebra.   Much of the work in Mathematical Logic in the 100 years and more that have gone by after The Laws of Thought has been given to taking out the errors from Boo1e's ideas, making some of the parts of his theory stronger, putting his algebra in the form of a system of deductions and, lastly, moving from it towards more general theories in Abstract Algebra, for example towards the theory of those ' part-ordered' classes every two elements of which have a greatest lower limit and a smallest higher limit, such part-ordered classes being what are named lattices. We will at this stage say something about one or two of the ways in which Boole's algebra was dnnged while he was still living or not long after his dalfhg these changes were made by Ievons, Peirce and Schroeder.
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7.2. The work of Jevons.   William Stanley Jovorns (1835 - 1882) was at the same time head of the school of logic and philosophy and head of the school of economics at Owens College, Manchester (later the Univesity of Manchester), from 1866 to 1876, going then, because of his heavy teaching work, to University College, London, as head of the school of economics. Like other men of science or letters under the rule of Victoria (for example Darwin and Clifford, Browning and Carlyle), Jevons was often ill, having frequent pains in the head and sleeping badly. So in 1880, when he was only 45 years old, he gave up his position at University College, living after that on his private income and the money from his books. But death came to him suddenly two years later while he was having a swim at Hastings. This event put an end to the writing of a full account of his teachings on economics. However, what had been produced by him earlier in this field had made his the chief name in current English economics. In addition, his was the chief name in the current English logic of deduction and logic of natural science ; this was the effect of his six high quality books on logic.
    The first of these books was Pure Logic, of the Logic of Quality apart from Quantity (1864); the greatest of them was The Principles of Science (1874) in which the writer's views on the logic of deduction and the logic of natural science were put, together, in their complete form.
   One of Jevons's interests was in ' reasoning machines', of which he was the first to make one. Much of his time in the 1860's was given to the building of such an instrument by which the validity of deductions was able to be tested autornadcaily.
   In Jevons's opinion the theory of deduction is a theory of reasoning about qualities or quality-ideas and is not one of statements as such or of classes. The theory of deduction has to be one under whose rules the - 45 - steps of any reasoning are turned into equations the signs on the two sides of which are representatives of clearly marked out qualities ; certainly not turned into classes like the uncertain classes of which Boole made regular use in The Laws of Thought. ' All horses are animals' would be seen by Boole in The Laws of Thought, as it would have been seen by Hamilton, as saying 'all horses are some animals' ; Boole's equation for this would be h=v x a, h being the class of horses and a the class of animals, while v is a class unfixed in every way but this, that some of its elements are animals. But which of its elements are aninuals ? Jevons's answer is that it is those which are horses. The right equation for ' all horses are animals' is h=h x a, where ' h ' and ' h x a ' are names of qualities, one quality that is simple and one that is complex.
    Jevons's theory was different from Boole's in another point. He had the idea that the heart of deduction is in the exchanging of signs, of things that are equal to one another, for one another. In his words, ' whatever is true of a thing is true of its like'. This idea was important later when one came to see that it is necessary to say what rules there are in a system by which deductions may be made. In the time of Boole and Jevons and for long after -- till the early 1900's -- no one, but for Frege, was conscious of the need to keep the laws in a system separate from its rules of deduction, which are for going from given laws to new ones. - 46 -

The following is uneditied !!
Chapters 10, 11 & 12 are okay.



7.3. Comparison of Boole's and Jevons's systems.   It was Jevons's belief that Boole went wrong in looking at as a branch of Mathematical Analysis and that his errors were the outcome of this. Every line in the inciting out of a question in logic has to have a sense from the point of view of logic. Boole lets a great number of lines be used though they have no such sense. Further, by Boole's account of the operation of addition (+) of classes the operation may be done only on classes which have no common elements, however, by his rules he has to let ' x+x ' and ' x+(x+y) ' be parts of equations though x and x, and x and x-1-y, have common elements (if x and y are not Nothing). Again, while it is a general law of Boole's algebra that xx (y+8)= (x x y)-l-(xxa), it is possible for only one side of this equation to be well-formed, given Boole's account of +, because sometimes it is possible for the operation of addition to be done on xxy and xxx, these two classes having no common elements, while it is not possible for the operation to be done on y and z,these two classes having common elements, as when 2 and 3 are the elements of x, 2, 4 and 5 are the elements of y and 3, 4 and 5 are the elements of z. Jevons put forward with much force the suggestion, which had been made before by De Morgan in connection with classes, that it would be better if the operation of addidon in logic were made to be one for which there is no need of the things on which the operation is to be done to be completely different and to have nothing whatever common to them. In class algebra x-4-y would then be the class of all the things that are elements of x or of y or of x and of y. One value of this suggestion is its causing number signs such as 2x(=x+x), which have no 8ense in logic, to be droppedout of the algebra, x+x now being equal to x. With this change of addition Boole's operation of taking away (-) is no longer the opposite of + and i~ no longer like the ._ of arithmetic or number algebra ; -x is made to be the class of all the things that are elements of All but are not elements of x, so now, for every class x, it will be true that x-4--x= 1, 1 being All, unlike in number mathematics where .¢+-x=0.
- 47 - Another value of Jevons's suggestion is that De Morgan's Laws, put forward by De Morgan for classes, these Laws being that -(x+y)=-xx -y and -{xxy)=--x+ -y, become true in the new theory, and this is a help in the answering of questions about reasoning. Though ]evons's theory of deduction was better in some ways than Boole's, it was generally less good and more complex. The effect of this was that only two of his ideas were taken into the body of Mathematical Logic by later workers. One was the idea of changing the operation of addition. The other was his way of making an expansion of an equation to take more qualities or classes into account; for example, an expansion of x=x is possible by which y is taken into account, as in x=x=xx(y+-y)=(xxy)-I-(xx -y). Such expansions were important for the later development of the idea of what was named the ' normal form ' of a complex of 8igns, the normal form being a fixed way of writing any such complex so that comparisons between com- plexes and decisions of certain sorts about their proper- des may be readily made.

7.4. Peirce's points of agreeement with Boole and Jevons.   Charles Sanders Peirce (1839-1914) was a teacher of philosophy, in logic, for only 5 years, at Iohns Hopkins University 'm 1879-1884. He made his living for 28 years, from the time he got his degree from Harvard till 1887 when he was 48, from a branch of United States government work in science; after 1887 he had no regular income and was often so poor and in debt as not to have enough money for food and heating. His ideas on Mathematical Logic were given in papers, 011 recorded in private notebooks, whose writing wok place at widely different times between about 1866 and 1905, the most important years being those when - 48 - he was at Johns Hopkins. Unlike Jevons, Peirce gave his approval to Boole's view of the logic of deduction as having close connections with mathematics, though he was in agreement with Ievons that any system of such a logic has to be limited in its use of mathematics by the need of letting adl its laws and rules and all the steps which may be taken in it have good sense in logic. There are four parts of logic that were much helped to go forward by Peirce : the logic of classes, of relations, of statement connections and of statements in which there are variables. These four will be touched on in turn.

7.5. Peirce's use if the ' a part of ' relations.   Peirce was the first worker in Boole's sort of logic to make use of the new operation of addition that had been put forward by De Morgan and Ievons. Down the years from 1867, the year ofhis first papers on Mathematical Logic, Peirce was responsible for' a number of important developments and of new systems in the algebra of classes. Specially to be noted was his joining a new relatiOn between classes to those which there had been in Boole's algebra. This new relation was 'a part of', which is currently marked by the sign C ; the sense of x C y is that every element of the class x is an element of the class y. In Boole's algebra thought is able to be given to this relation, which is one of.the most important ones in logic and mathematics, only by an equation : xx(l-'Y)=0. But it is in fact a great help in logic and mathematics to have a simple sign for this relation. Further, there is this to be said for Peiroe's use of the 'a part of' relation as a simple relation: this relation in the algebra of logic is like the relation of 'less than or equal to' in number algebra and, keeping this in mind, one may make new discoveries of laws of logic. If the - 49 - algebra of classes is, in its form, like the algebra of the numbers 0 and 1, then, because there are true laws ming the relation 'less than or equal to' in this number algebra, there are true laws using the relation 'a part of' in the class algebra, This was overlooked by Boole. Lastly, the sign C has the sense in the algebra of statement connections of material implication (2.8); when there was a separate sign for this statement connection, one was more readily able to give it attention and to make discovery of its properties. Peirce himself and others gave it much attention. Because of this, it became the generally used statement connection of the implication sort almost without a iight, even though material implication is a somewhat special and strange sort of implication. Its attractions are that it is the least strong sort of implication and that it is generally better than the others for the purposes of mathematics. - 50 -

7.6. Peirce's theory of relations.   Peirce gave a full account of his theory of relations in his paper 'The Logic of Relatives' printed in Studies in Logic by Members of the Johns Hopkins University (1883). His theory is based on Boole's algebra and on De Morgan's Formal Logic (1847) and papers of De Morgan. De Morgan put much weight on the idea of relations such as 'to the right of' which if they are true of the things a and b (in that order) and are true of the things b and c (in that order), then they are true every time of the things a and c (in that order): if a is to the right of b and b is to the light of c, then a is to the right of c. Further, De Morgan gave weight to the idea of an operation of joining two things together in thought so that they have a relation between them such as that between a head and a house in what is marked by ' a head of a horse' or between a daughter and a teacher in what is marked by ' a daughter of a teacher'. (It may be pointed out that there are in fact a great number of different relations of this sort, though this is kept dark by the uncertain connection word ' of '.) It was De Morgan's belief that the ' of ' relation was in some ways like the operation or relation marked by x in arithmetic or algebra ; an example is the number law a x (b+c)=(a x b)+(a x c), which is, in addition, a law in the logic of relations, giving x the sense of ' of ' and + the sense of ' or ' : the father of a teacher or a manager is the father of a teacher or the father of a manager. However, though relations were talked about by De Morgan and what he said had no small effect on Peirce, he did not get so far as building a systemlor a mathematics of relations; this is what Peirce made the attempt to do.
    In 'The Logic of Relatives' an attempt was made by Peirce, using the work of Boole and De Morgan, to give a general theory like a mathematics of things having relations, these wi]1 be named here 'relation things'. The relation thing 'father' among persons may be said simply to be the class of all the groupings (I : J ) where I is a father and .I is an offspring of I; so (James Mill : John Stuart Mill) is an element of the class that is the relation thing ' father ' but (Roger Bacon : Francis Bacon) is not an element of that class. More generally, any relation thing r may be said to be the class whose elements are all the groupings (I :J ) of the things I and J' such that, as one would commonly say, I is an r of J (or something near this). One value of this account by Peirce is its putting an end to the belief that a relation is something very strange and special. In this way Peirce did for relations what Boole had done for possible subjects and predicates' as qualities had been turned by Boole into classes of the things that have the qualities, relations were turned by Peirce into classes of 51 the groups of things between which there are the relations. Another value is in its suggestion that ideas in mathematics which might be viewed at first as not a part of logic at all are able to be given definitions which nnake use only of what is a part of logic ; mathematics and logic then are not quite as separate as it seemed, and it might even be possible for all the ideas of mathematics to be broken up into ones of logic, mathematics becoming an expansion of logic and no more than an expansion of it. (A definition is a statement or rule saying what sense some word or other sign has or is to have, or saying that some simple or complex sign is going to be used as equal to another sign or group of signs.) - 52 -

7.7. Peirce's logic of statement connections.   A logic of relations is needed for a logic of mathematics because there is a great number of relations playing an important part in mathematics, for example 'square root of', ' between', ' in the same plane as', ' greater than', ' equal to'. Relations are some of the material of mathematics. But the logic of statement connections is a theory of a part of the form, not of the material, of mathematics. Not that the business of these two sides of logic is only with mathematics -- far from it. But much of their development, and that of other sides of logic, has in fact been caused by the desire of building a logic that is at least a good one for the needs and purposes of mathematics. There were chiefly three new ideas in the logic of statement connections for which Pdroe was responsible. The first was the idea that it is possible to give definitions of all the different statement connections between one or two statements, as ' not', ' and', ' or' and the material implication ' if ... then', by ming the one connection 'the two are false' or by using the one connection 'not the two are true'. For example, ' the two are False' being marked by the writing of the statements s and t side by side so that 's and t are false' becomes ' st', then ' s is false' becomes ' ss', 's is true or t is true' becomes ' (st) (st)', and so on.
    Secondly, Peirce gave an account of a ' decision process' for testing if a complex of signs in a system of statement connections is or is not representative of a true law of logic. A process is said to be a decision process if it is fully automatic -- able to be done by machine -- and gives a test of which complexes of signs in a system are laws of the system, a complex of signs being a law if and only if it has a certain property P. In Peirce's logic of statement connections the decision process is to give the values 'v and f (from the Latin. verum = true and falmm = false) in all possible ways to the variables in the complex of signs and, by definitions, to get the value of the complex itself worked out from the values of its parts; if this value is necessarily v for all the possible values of the variables, then the complex is a law, while if this value may be f for some values of the variables, the complex is not a law; so the property P here is the property of a complex's having the value v for all possible values of the variables in it. Peirce makes this decision process clear in relation to the complex C that is of the form (s->t)->[(t->u)-> (s->u)], where -> is a sign of material implication. By definition, A->B has the value U whatever values v and fthe parts A and B of a complex might have but for A having the value o and B having the valuef, when A->B has the vaduef: a material implication is false if and only if the condition is true and the outcome is false. Now Peirce's argument is this, that if C ever has the value f, then, from the definition, s->t has the value v and (t->u)~>(s->u) has the value f; but if this last has the
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value f, then, from the defiinition, t->u has the value v and s->u has the value f; and 80, again from the definition, s has the value v and u has the value f. Because u has the value f and t->u has the value v, necessarily, from the given deinition of material implication, r has the value But then, because s has the value v and 11 has the value f, s->t necessarily has the value f. This, however, is not in harmony with s->t having the value U when C has the value f (10 lines back). So it is not possible for C to have the value f at all' if it is given that value, one gets values that are not in harmony with the definition of material implication. In ways like this all complexes of signs in a system of statement connections may be tested to see if they may be given the value f without eifecting values that are not in harmony with some definition, in relation to v and f, of a statement connection.
    Thirdly, Peirce had the idea of building upa theory of the logic of statement connections as a system based on axioms. Because the deduction of theorems about statement connections from axioms is very hard, it was Peiroe's suggestion that the decision process we were outlining a minute back be uSed for supporting the right of those complexes which are unable to have and to keep the value f to be theorems in the system and so representatives of laws of the logic of statement con- nections. The validity of any argument in mathematics which is dependent on statement connections is con- ditioned by these laws: if the argumentS validity is dependent only on statement connections, then the argument has validity if and only if it is in agreement with these laws; if the argument's validity is dependent on statement Connections and on other sorts of reasoning, then agreement with those laws is necessary but not enough to make certain of validity. Peirce's system has four axioms in which material implication is the only statement connection and one axiom whose purpose seems to be to give a definition of the idea of ' not ' or ' false '. - 54 -

7.8. Peirce's quantifier logic.   The statement ' 4 is the square of 2 ' is true or false, as is ' the death of Newton took place in 1900'. We will say that such statements are two-valued ones, the values being 'true' and 'faise'. On the other hand, 'x is the square of y', where the range of the variables is the class of the numbers 1, 2, 3, ... , and 'the death of x took place in y', where the range of x is the class of persons and the range of y is the class of years in the current system, are statements that are not two-valued ; they are not true and they are not false. We will say that a statement which is not two- valued and in which there are variables is a statement with free variables. s(x), t(x, y), ufx, y, 2) and so on are used for noting such statements, for example s(x) is the name of any statement having one variable in it. (This is a little over-simple; it is unnecessary here to go into delicate details, because we are having only a bird's-eye view of logic.)
    Looking at the statements 'some x is the father of some y', 'some x is the father of every y', 'every x is the father of some y' and 'every x is the father of every y', the range of the variables being the class of persons, one sees that all four of them are two-valued; the first is true and the others are false The first statement says that someone is somebody's father, the second that somebody is the father of everyone, the third that everyone is somebody's father and the fourth that everyone is the father of everyone. Putting these four statements in a somewhat different form the iirst be- comes 'there is an x and diere is a y such that: x is the - 55 - father of y', the second becomes 'there is an x such that for every y: x is the father of y', the third 'for every x there is a y such that: x is the father of y' andthe fourth 'for every x and for every y: x is the father Ofy'. Though the part coming after the : in these four state- ments is a statement with free variables, the complete statements themselves are two-valued, 'There is a' and 'for every' are named quaotyiers, the first being the 'existence' and the second the 'general' quantiiier, » which were noted by Peirce by the Greek letters Z andII. Peirce gave much attention to the logic of variables, this being that field of logic whose business is with implications and arguments some or all of whose con- ditions or outcomes are statements with free variables or that have quantifiers in them. A great number of theorems and arguments in mathematics are dependent on the use of quantifiers; so this part of logic is very important. With Frege but having no knowledge of Frege's work, Peirce was the first to be interested and to make deep discoveries in this held. Using - as the sign of 'not' or 'it is false that', some simple examples of laws given by Peirce are : and laws which are the same as (3) and (4) but in which x takes the place of +. - 56 -

7.9. Schroeder's algebra of logic.   Ernst Schroeder's (1841 - 1902) two chief books on Mathematical Logic were Der Operationshreis der Logikkalkus ('The Range of Operations of the Mathematics of Logic') and Vorlesungen über die Algebra der Logik (' Teachings on the Algebra of Logic'). The first of these two books (1877) is very short, being only 37 pages long. There is in it a simple system of the algebra of logic, this system being based on Boole's though different from it in some important points where the discovery had been made by later thought and experience that changes were wise or necessary. The tendency of these changes was in the same direction as that of those changes that had been put forward by Ievons and Peirce: the algebra is to be logic in the dress of mathematics, not a bit of mathematics which almost by accident is open, completely or in part, to being viewed as a theory of logic.
    The writing of the Vorlesungen was in four parts that came separately on the market between the years 1890 and 1905. This book gave a full account of the current Mathematical Logic in so far as this had its roots in algebra, in it were discussions and systems going into the smallest details of the logic of classes, of statement connections and of relations.
    Much that Schroeder did was new, though often resting on older material. Among the number of his theorems that were of special value was that to the effect that all the laws of the algebra of logic may be grouped in twos, it being possible from a knowledge of One of the two to get straight away, by a simple rule of exchange, a knowledge of the other one. The ruie is that, in the given law, + and x , and 1 and 0 (the signs of All and Nothing), be put in one another's places; for example, from the law x x 1=x one may get the law x+0=x, from the law x+-x=1 one may get the law xx-x=0 and from the law xx(y+z)=(xxy)+(xxz) one may get the law x (yx z)={x y)x (#=+#) This last law was first given by Peirce; it is not a law in Boole's - 57 - algebra because of the different use Boole makes of +- Again, the laws x+x=x, x+yIy+x, (x+y)+z=x+ (y-I-af) and x=x+(x xy) and the four laws that may be got from these by the rule of exchange were seen by Schroeder to be of a somewhat separate sort from the other laws, In fact, these 8 laws are very important for Lattice Theory (see end of 7.1) because a class L is a lattice if and only if these laws are true in it.
    Not to be confused with Schrödinger's cat, a paradox devised by Erwin Schrödinger in1935.

7.10. Whitehead's and Huntington's work on the algebra of logic.   The present history of the algebra of logic will be ended now by our saying a word or two about the stage in the development of this algebra that came 'm the ten years after its first 50 years, 18471891 This stage is marked by two things. Firstly, by Whitehead's (1861-1947) attempt in his Universal Algebra (1898) to give a completely general theory of algebra and to say what bodies of axioms and definitions will be good starting points for the different systems covering this part of mathematics ; this book was the first after The Laws of Thought to take seriously the algebra of logic as a field of everyday mathematics and it was the Erst to make clear the connections between this branch and other branches of Abstract Algebra. Secondly, this stage is marked by the Erst conscious work in the metamathematics of the algebra of logic (see 51). Using tricks and processes of a sort designed by Plano and by others of the Italian school, Huntington (1874-1952) gave a demonstration -- a good argument in support of what is said to be true -- of the consistency of a number of different axiom systems of the algebra of logic, the substance of these systems being copies of ones in the books of Schroeder and Whitehead, and he gave a demonstration that every axiom of his systems is unable to be got as a theorem from the rest of the axioms of the same system, so no axiom of a system is dependent on`the others. This bit of second-order theory of the algebra of logic -- the logic of logic -- was done at the time (1904), between 1890 and 1905, when second-order theories of mathematics --- metamathematics -- were starting to be seen as interesting and as a very important part of the theory of the bases of mathematics .
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8   FREGE'S LOGIC

8.1 . The mathematics of logic and the logic of mathematics.   Mathematical Logic has two sides. One, in its roots in the past (we say this because all sciences have a tendency to become self-conscious and, if necessary for their best development, to let the material that was their first business get dropped from view), is the science of the deduction form of reasoning, this science using ideas and operations like those in mathematics and copying mathematics in being system- making. This side, then, is the mathematics of logic. It was in this that Boole and others interested in the algebra of logic were working. This side has two levels : the logic of deduction itself in the form of systems covering statement connections, classes, relations and statements with variables, and the second-order theories of those systems, whose birth came nearly 50 years after that of those first-order systems. The other side of Mathematical Logic is the logic of mathematics. Here there are two chief sorts of questions to which one - 59 - has given attention, certain points of these being in the same line of direction. One sort of question is that to be answered by the metamathematics of any given branch of mathematics--of arithmetic, Euclid's geometry, group theory or whatever it may be. The second sort of question is about the bases of mathematics; for example, is it possible to have a system with the property of consistency in which the ideas used are enough and have the right properties so that, firstly, definitions of all the ideas of the common mathematics may be given from these and only these ideas, and, secondly, all the theorems in the common mathematics may be got as theorems from that system stretched to have new definitions resting on the ideas in the old ones ? From the time of Schroeder's death (1902) till now, most work in Mathematical Logic has been in connection with the logic of mathematics, and in 8.2-12.6 we will say something about this work. The reading of this coming discussion will have to be done very slowly if the material is going to be taken in. Because we have little space and because it is possible to get a clear view of what has been produced by the newer work in Mathematical Logic only if one goes through the long stretch of expert details, we will be limiting ourselves to outlining some of the chief directions which the later developments in Mathematical Logic have had. - 60 -

8.2 Reasons why Frege's writings were undervalued.   Gottlob Frege (1848-1925) was Professor of Mathematics at the University of jena and the writer of books and papers on logic and the logic of arithmetic, and on the philosophy of these fields. For a number of reasons Frege's writings were undervalued till Russell's Principles of Mathematics (1903) gave an account of them. One reason was their use of signs for statement connections the writing of which was not in the normal direction of left to right but up and down; this makes the reading of sign complexes much harder. Another reason was that, generally, those persons whose business was philosophy were controlled by the impulse not to go near, still less to go through, a mass of sign complexes when the purpose of this mass was to be the support of a theory in philosophy -- here, in the philosophy of mathematics. Their belief was (and is) that an argument for or against any such theory has to be made in everyday language because the value of the theory is dependent on its starting points, and the opinions which are at the back of these, being true, and that it is not possible for these to be supported by anything inside a mathematics-like structure but only by arguments outside t he structure. Another reason was that Frege's theory was so surprising and so far from probable that there seemed little chance of the trouble of working through the details of the deductions offered for it being rewarding. One last reason we would put forward is that what Frege was doing was off the straight and narrow road: it was not quite normal mathematics, it was not quite normal philosophy and it was not quite normal logic. It was not normal mathematics: it was the bases of mathematics. It was not normal philosophy : it was the philosophy of mathematics done by someone having a first-hand knowledge of mathematics. And it was not normal logic: it was not the algebra of logic. (The 'normal' is in relation to the time 1875-1900.) - 61 -

8.3 Frege's view of the relation between logic and arithmetic.  What was Frege's theory? It was that number is an idea (not dependent on the mind for its existence) which may be broken up into ideas of logic and which is open to a complete definition by using no - 61 - more than these ideas ; and so, further, that arithmetic itself is a part of logic. In agreement with Frege, Russell said that arithmetic ' is only a development, without new axioms, of a certain branch of general logic'. The journey to the full, tightly reasoned demonstration of this theory was started by Frege in his Begriffsschrift ; eine der arithmetischen nachgebildete Formelsprache des rm|en Denkens ('Idea-Writing; a sign language, copying arithmetic, of thought as such') of 1879 and came to its end with the two parts of his Die Grundgesetze der Afithmetik, begrzfsschriftlich abgeleitet ('The Root Laws of Arithmetic, got by deductions in idea- writing') of 1893 and 1903. In the Begriffsschrift Frege gives an axiom system of the logic of statement connections and, using the general quantifer (by which, together with ' not ', he gives a definition of the existence quantifier), he goes deeply into the logic of statements with variables. In addition, there is an account of some parts of arithmetic worked out from definitions based on what are taken to be ideas of logic, for example the ideas of class, class element and relation, In the Grund- gesetze there is a complete account of arithmetic, in so far as this is the theory of the laws of the operations on numbers of the form p|q where the range of q is +/-l, +/-2, +/-3, ... and the range of P is the same together with 0. The heart of Frege's arithmetic is his definition of a cardinal number. The simplest cardinal numbers are 0, 1, 2, 3, ... For Frege, a cardinal number is a property of a class. To the question, 'What is the number of men in the Russian army now' the right answer will be a cardinal number and this number will be a property of the class 'man in the Russian army now'. By a 'class' here Frege, unlike Russell, has in mind the class-idea (that is, the class as an idea), not the list of things coming under the class-idea. So a
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cardinal number is a property of a class-idea, for example of the class-idea ' man in the Russian army now '. Flege's definition of the cardinal number of a class x is that it is the class of all the classes y such that x and y have a one-one relation between them; by definition, classes x and y have a one-one relation between them if and only if (i) each element of x may be grouped with some element of y, (ii) each element of y may be grouped with some element of x, (iii) if any a1 and a2 of x are grouped with a certain b of y, then a1 and a2 are the very same element of x, and (iv) if a certain a of x is grouped with bl and with b., of y, then b1 and B2, are the very same element of y. This may seem harder than it is; it is put like this for the purpose of stopping anyone from being able to say that the definition goes round in a circle, because it makes use of the cardinal number one (in 'one-one relation'). What the definition says is simply this, that x and y have a one-one relation between them if and only if it is possible for there to be a body R of groupings (a : b) such that every a of x is the first name in some grouping (a : b) of R while no a of x is the first name in more than one such grouping and every b of y is the second name in some grouping (a : b) of R while no b of y is the second name in more than one such grouping. From his definition of cardinal number and by the instruments of logic Frege gets the theorems of everyday arithmetic at the end of very long chains of deduction in the form of sign complexes; so even if one has doubts of his theory being true, at any rate his industry and expert invention are to be respected. - 63 -

8.4 Frege on functions.   One of the most important ideas at the base of mathematics is that of a function. The definitions given of this idea in Frege's time did not, when looked at with care, make very good sense though they had seemed to make good sense and, further, they were not in harmony with the uses made of the idea. As frequently in philosophy, Frege had a sharp tongue when attacking, specially when attacking an unsafe position.
    What is a function ? In Frege's time the common answer to this question was to say, as the German writer on mathematics, Ernst Czuber, did : 'If every value of the number variable x that is an element of the range of x is joined by some fixed relation to a certain number y, then, generally, y, like x, comes under the definition of a variable and y is said to be a function of the number variable x. The relation between x and y is marked by an equation of the form y f(x).' (Number variables are not necessarily limited to the range 0, :l:1, :t2, i3, . . ., or even to numbers of the sort p/q where P and q are from that range (q-=0): they may, for example, be variables of numbers like ~/2 or *s/-25.) This answer was soon undermined by Frege's blows, one after another, from the gun of his quick-firing brain. 'One may straight away take note of the fact', says Frege, 'that y is named a certain number, while, on the other band, being a variable, it would have to be an uncertain number. y is not a certain and it is not an uncertain number; but the sign 'y' is wrongly placed in connection with a class of numbers [the range of the variable y] and then it is talked about as if there were only one number in this class, It would be simpler and clearer to say: "With every number of an x-range there is some fixed relation by which it is joined to another [not necessarily different] number. The class of all these last numbers will be named the y-range." Here we certainly have a y-range, but we have no y of which one might say that it was a function of the number variable x.
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    ' Now the limiting of the range does not seem to have anything to do with the question of what the function itself is. Why not take as the range the class of all the numbers of arithmetic or the class of all the complex numbers (of which the first class is a part)? The heart of the business is in fact in quite a different place which is kept out of view in the words "joined by some fixed relation". What is the test of the number 5 being joined by some fixed relation to the number 4? The question is unable to he answered if it is not somehow made more complete. With Herr Czuber's account it seems as if, for any two numbers, the decision was made at the start that the first is, or is not, joined by the relation to the second. Happily Herr Czuber goes on to say : "My definition makes no statement about the law controlling the joining by the relation; this law is marked in its most general form by the letterf but there is no end to the number of different ways in which it may he fixed." Joining by a relation, then, takes place in agreement with some law and it is possible, by the use of thought, for different laws of this sort to be produced. But now the group of words "y is a function of x" has no sense, there is need of its being made complete by the addition of the law. This is an error in the definition. And, without doubt, the law, which is overlooked by this definition, is truly the chief thing. We see now that what is variable [changing] has gone completely out of view and in its place the general has come into view, because that is what is pointed at by the word "law".' - 65 -

8.5 Functions as rules.   What Frege says about functions in the paper from which we have taken these lines has become normal in the present-day account of the idea of a function. A function f is a rule about classes X and T, X being named the class of definition off and T the range off, the rule producing a class of ordered groupings (x : y) where x (which is named an 'argument' of f) is an element of X and y (which is named the 'va1ue' of f for x) is that selection from among the elements of T which is made in agreement with the rule ; the class of all the selections y from T is named the class of values off For example, if the class of definition of f is the class X of all the numbers 1, 2, 3, . . ., the range of f1 is the same class and f is the rule ' To every element x of X let 2x be joined', then the class of ordered groupings produced by this rule has (1 : 2), (2 :4), (3 : 6), . . . as its elements, so the class of values of f has 2, 4, 6, . . . as its elements , again, if the class of definition of f, is the class X of all the numbers -1, -2, -3, . . ., the range of j, is the class of all the numbers 1, 2, 3, . . . and fl is the rule ' To every element x of X let x2 be joined ', then the class of ordered groupings produced by this rule has (-1 : 1), (-2 : 4), (-3 : 9), . . . as its elements and the class of values off, has 1, 4, 9, . . . as its elements .
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9   CANTOR'S ARITHMETIC OF CLASSES

9.1 Finite and enumerable classes.   A short way of writing that a one-one relation is possible between any two classes x and y is 'x ~= y'. If x ~= y, then by a definition of Cantor's (1845-1918), x and y are said to have the same cardinal number or the same power, the sign - 66 - for this fact being the equation'§=§'. (This definition is clearly like Frege's definition of ' cardinal number'; but Cantor does not give a definition of ' cardinal number', only of ' having the same cardinal number'. Cantor and Frege were working at the same time, but quite separately; one was not copying the other at all.) If N,,1 is the class of the 'natural' numbers from 1 through m, so the elements of it are 1, 2, 3, ... , m--1, m, then a class x is finite if and only if it has no elements or, for some natural number m, x- N,,,. If N is the class of all the natural numbers, its elements being 1, 2, 3, , .. where '.. ,' is used for marking that the list goes on for ever, then a class x is said to be enumerable if and only if x ~= N. It is strange how great a number of quite different classes are enumerable. For example, Cantor gave demonstrations that the class Z of all the numbers 0, il, :t2, :l;3, ... is enumerable; the class of all the numbers p/q greater than 0 and less than 1., where P and q are elements of N, is enumerable; but what to the mind's eye would be seen as a very much greater class, the class Q of all the numbers p/q, where P and q are elements of Z (q-=0), is again enumerable ; and, further, the class of all numbers which are roots of an equation of the form a,,w"+a"_1w"-1+... +a1w+a0,0, where each "4 is an element of Q, is enumerable.
    In view of these surprising facts it is natural to put the question: Are there any classes that are unenumerable, that is which are not finite and which are unable to he placed in a one-one relation with N? The answer is, Yes. The class I of all the numbers of the form 0-a1a,a: ... an ..., where the a's have values in the range 0, 1, 2, ..., 9, is an unenumerable class, it has, for example, 0-127845 '7 023640000 4 l I and 0~765976 l n l as elements. There are other unenumerable classes.
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9.2 Cantor's cardinal arithmetic.   Though Cantor did not give a definition of 'cardinal number' he made me of cardinal numbers as things having some sort of existence and having properties and relations among themselves, in much the same way as one makes use of the numbers 1, 2, 3, ... in everyday arithmetic without definitions of them. By definition, the cardinal number of a class x is greater than the cardinal number of a class y -if >§-if and only if there is some part x1 of x such that X1 is not all of x, xlzy and there is no part y1 of y such that yl ~= x, for example, N, >N, and N> Nina- Cantor gave a demonstration of a general theorem which says that, for any class y having elements, the class Uy whose elements are all the classes x such that x Cy (7.5) has a greater cardinal number than y has: Uy >y. An outcome of this theorem is that there are ever greater and greater cardinal numbers, if only because y < Uy < UUy < UUUy < UUUUy < ...
    An arithmetic of cardinal numbers was worked out by Cantor, definitions of the operations of addition and so on for cardinal numbers being given and the discovery and demonstration made of laws of the arithmetic. To take the simplest operation, that of addition : by definition, if x and y are any two classes such that no element of the one is an element of the other, then 219 = 3:+5', where the + sign in the left-hand side of the equation is of the addition of classes (x +y being the class whose elements are all the things that are elements (of x or of y). The common laws of addition are true for cardiinal numbers ; if k1, k, and k, are cardinal numbers, k,+k,==k,+k, and (k,+k,)+k,=k1+(k,+k,). On the other hand, there are, further, some uncommon laws ; - 68 - for example, if m is the cardinal number of any finite class and k is the cardinal number of N, then the addition of m k's is equal to only k (mk=k) and km=k.

9.3 Cantor's ordinal arithmetic.   The arithmetic of still another sort of number, that named cardinal, was worked out for the first time by Cantor. Ordinal numbers are based on well-ordered classes, that is on classes which are made to have a certain order among their elements so that there is a first element, a second element, and so on. For finite classes ordinal and cardinal numbers are the same; however, for classes that are not finite the numbers are not the same because it is possible to make all such classes have widely different order relations. And, as for cardinals, there is no limit to the number of higher and higher ordinals. The theory of cardinal and ordinal numbers -- the arithmetic of classes -- has been one of the most fertile inventions of the newer mathematics and is one of the most important branches of Mathematical Logic. The details of this theory are highly complex, and we are unable to go into them here.

9.4 Burali-Forti's and Russell's discoveries.   But the warm bright days when the sunlight was playing on the flowers in Cantor's garden and all seemed well were sadly soon over. Black clouds came up in the sky and the earth was made dark. Men of mathematics were put in doubt, fearing to see the downfall and destruction of one of the beautiful theories of mathematics. The cause of the trouble was the discovery that Cantor's arithmetic of cardinal and ordinal numbers did not have the needed property of consistency (5.1). In 1897 Burali-Forti (1861-1931) gave a demonstration
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that the ordina1 number of the well-ordered class W of all the ordinal numbers is not an element of that class ; so, on the one hand, all ordinal numbers are, by definition, elements of W while, on the other hand, not all ordinal numbers are elements of W because the ordinal number of W itself is not an element of W. Again, in 1901 Russell (1872-) saw the loss of consistency of cardinal arithmetic in connection with the most general class A. By definition, A is the class whose elements are all the things which are not classes, together with all the classes of those things, together with all the classes of classes of those things, and so on. A, then, is a class such that all its parts are elements of it. Because of this fact, it is not possible for there to be more parts of A that there are elements of A, that is MA (the cardinal number of the class of all the parts of A is less than or equal to the cardinal number of A). However, this is not in agreement with Cantor's general theorem (9.2) which makes it necessary here that the cardinal number of UA is not less than or equal to, but greater than, the cardinal number of A. From the discovery of this example Russell got the idea of another one which is the most noted of all such examples. He sent it to Frege at the very time that the second part of Frege's Grundgesetze der Afithmetik was coming near the end of its printing before being offered to the public ; Frege was made conscious of the fact that his system of classes, like Cantor's, did not have the needed property of consistency. Russe1l's new idea was this : without there being any suggestion that any class is able to be an element of itself, let E, by definition, be the class of all classes that are not elements of themselves , for example, the class ' man' is an element of E because ' man' is not an element of itself, this class having only men, and no classes, as elements. Is E an
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element of E ? If (I) E is an element of E, then (II) F is not an element of E because the only elements of E are those classes which are not elements of themselves, on the other hand, if (II) E is not an element of E, then (I) E is an element of E because of the definition of E. By a common law of logic one or the other of (I) and (II) is true. But whichever one is true makes the other one true in addition. So the theory of classes has two Opposite theorems in it : E is an element of E ; F is not an element of E.
    The effect of these discoveries on the development of Mathematical Logic has been very great. The fear that the current systems of mathematics might not have consistency has been chiefly responsible for the change in the direction of Mathematical Logic towards metamathematics, for the purpose of becoming free from the disease of doubting if mathematics is resting on a solid base. A special reason for being troubled is that the theory of classes is used in all other parts of mathematics, so if it is wrong in some way, they are possibly in error. Further, quite separately from the theory of classes, might not discoveries of opposite theorems in algebra, geometry or Mathematical Analysis suddenly come into view, as the discoveries of Burali-Forti and Russell had done ? It has been seen that common sense is not good enough as a lighthouse for keeping one safe from being broken against the overhanging slopes of sharp logic. To become certain with good reason that the systems of mathematics are all right it is necessary for the details of their structures to be looked at with care and for demonstrations to be given that, with those structures, consistency is present.
    This last point and the fears and troubled mind we have been talking about in these lines have been and are common among workers in what is named 'the bases of
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mathematics', that is axiom systems of logic-classes-and-arithmetic. However, some persons, with whom the present writer is in agreement, have a different opinion. They would say that the well-being of mathematics is not dependent on its 'bases'. The value of mathematics is in the fruits of its branches more than in its 'roots'; in the great number of surprising and interesting theorems of algebra, analysis, geometry, topology, theory of numbers and theory of chances more than in attempts to get a bit of arithmetic or topology as simply a development of logic itself. They would say that the name 'bases of mathematics' is a bad one in so far as it sends a wrong picture into one's mind of the relations between logic and higher mathematics. Higher mathematics is not resting on logic or formed from logic. They would say that the troubles in the theory of classes came from most special examples of classes, and such classes are not used in higher mathematics. They would say further that though to be certain of consistency is to be desired if such certain knowledge is possible to us, a knowledge of the consistency of the theories of mathematics that is probable is generally enough and the only sort of knowledge of consistency that one does in fact generally have. And they would say that such probable knowledge is well supported if the theories of mathematics have been worked out much and opposite theorems in them have not come to light. There is no suggestion in all this that Mathematical Logic is not an important part of mathematics; the view put forward is that there is much more to mathematics than Mathematical Logic is and might ever become. And there is no suggestion that the question of consistency and like questions, and the discovery of ways of answering them, are hot important; to no small degree Mathematical Logic now is as interesting and important as it is because of its interest in such questions and answers.
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10   PEANO'S LOGIC

10.1 Implication and mathematics.   In its birth Mathematical Logic was the theory of classes. The first person to have the opinion that the theory of statements (with or without variables) is more important was Hugh McColl (1837-1909) who in a group of papers on ' The calculus of equivalent statements ' (1878 and later) put forward the belief that the only business of logic is with the theory of statements and that the chief statement connection is some sort of implication. The thought that the theory of statements and not that of classes is the root of Mathematical Logic, and the thought that implication of one sort or another is the chief relation to be given attention in logic, soon became the normal beliefs of those at the head of this field. For example, Frege and Peirce were interested in the logic of statement connections as a branch of logic separate from the algebra of class logic and implication was specially important in their systems, Before Peano (1858-1932), however, no one made use of the logic of statements for making clear the arguments of everyday mathematics, and so viewing logic as an instrument for getting clear and tight reasoning in such mathematics. (Peano was the first to give the new logic the name of ' Mathematical Logic ', because of his view of it as an instrument for mathematics : for him, Mathematical
-73- Logic is the logic of mathematics.) And it had not been pointed out before Peano that implication is the chief relation in mathematics, all or almost all the statements that are true in any system of mathematics being implications. (It was because of this teaching of Peano's about the theorems of mathematics being implications that Russell, in the opening of his Principles of Mathematics, gave as his defnition of mathematics that this was the class of all statements of the form ' if s1, then s2 ', s1 and s2, being limited in certain ways.) And the idea that it was possible, by the use of logic, for all the statements in mathematics, and not only in arithmetic, to he put in the form of a language of made-up signs and for the demonstrations of all its theorems to be done by changes and exchanges of these signs, starting from axioms and definitions as sign complexes, had not been acted on in detail before Peano. The other experts in Mathematical Logic at that time were interested in logic for itself or were interested, as Frege was, in turning a bit of mathematics into logic ; unlike Peano they were not interested in the value of logic as an instrument for everyday mathematics as everyday mathematics (and not as logic or physics or philosophy).

10.2. Peano's purpose in his logic.   Peano's purpose in Mathematical Logic was to make the demonstrations in mathematics tight and free from loose reasoning. His view was that the value of what is offered as a demonstration and its being a good or a bad argument are not dependent on taste or inner feelings but on the argument's having the property of validity which is publicly testable. Peano said that because the old logic is not of much use in mathematics, the arguments in this not being syllogistic, it has been judged by some, Descartes among them, that what is clear to the mind as being true is the only test of an argument's being right. He gives the words of Duhamel : ' La déduction se fait par le sentiment de l'évidence, qui n'a besoin d'aucune régle, et ne peut étre supplée par aucune -- A deduction is made by a feeling of what is clear to the mind as being true, which has no need of any rule and is unable to be given by any.' Peano's views about demonstration were a reaction to views of this sort.

10.3 Some of Peano's discoveries and inventions.   To put the demonstrations of mathematics in a tightly reasoned form Peano undertook the discovery of all the ideas and laws of logic that are used in mathematics and undertook the invention of a body of signs for clearly noting these ideas and for the clear statement of these laws. Among his discoveries and inventions were these : (1) the definition of a class by a statement of the form ' the class of x's such that p(x)', this form being marked by Peano by 'xe px' ; (2) the idea that statements with free variables are different in important ways from two-valued statements ; (3) the use of full stops, in the place of signs such as (,) and [,], for grouping complexes of signs , for example, the sense of a v b.=.' . :~ a. ~b is ' "a or b" is equal to "not-(( not-a) and (not-b))" ' ; (4) the use, for the operations and relations of logic, of signs that are unlike those of mathematics where there might be danger of a wrong reading; (5) the first pointing out of the fact (which Schroeder and others had not been conscious of) that the relation of being an element of a class is very different from the relation of being a part of a class, Peano noting the first relation by e and the second by the 'horseshoe' Ↄ (which became the normal sign for material implication after its use for that purpose by Whitehead and Russell in their Principia Mathematica) ; (6) the idea of ' the so-and-so ', noted by ix, this idea being needed in connection with properties of which one has the desire to say that there is one and only one thing having them ; (7) the noting of the general quantifier (7.8) by writing the variables under and to the right of the statement connection, so that for example the sense of x e a. Ↄx.x e b is ' for every x, if x is an element of a, then x is an element of b '; (8) the noting of the existence. quantifier ' there is a ' by E printed back to front, that is by backwards-E. Most of these and no small number of other signs of Peano's became normal ones in Mathematical Logic after their use by Whitehead and Russell.

10.4. Peano's school of metamathematics.   More than logic itself is needed for getting tightly reasoned demonstrations in mathematics. It is necessary for there to be complete lists of axioms and definitions : there has to be a full, clear statement of one's starting points in any system. Peano did much important work in this field of axioms and definitions. But Peano's interest in this work was guided by questions of logic : metamathematics. He was interested in questions like this : What is the smallest number of ideas and axioms, and which ones are they, that are necessary for working out an axiom system covering such-and-such material ? And how is it possible to give reasoned answers to such questions?
    Round Peano in Italy was formed a company of those experts who were interested in the axiom bases of mathematics and in the use of a designed language of signs for the theorems and arguments of mathematics. This group headed by Peano had control of a paper Rivista di Matematica (1891-) and were joined in the writing of a book Le Formulaire de Mathématigues (first printing 1894-95, last printing 1908 when the language used in it other than the signs for the material of mathematics was Peano's international language Latino Sine Flexione) for the detailed development of their theories.
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11   WHITEHEAD AND RUSSELL'S PRINCIPIA MATHEMATICS

11.1 Short account of the three parts of 'Principia Mathematicai   The high-water mark of the first stages in the growth of Mathematical Logic, the stages from 1847 to 1910, was Whitehead and Russell's Princéria Mathematica (first part, 1910; second part, 1912; third part, 1913). This is a great book, great in quality and great in size -- its three parts have more thm 2,000 pages. In it are united into one general system the Boole-Schroeder algebra of logic and the theories, dilierently purposed from that and from one another, of Frege, Cantor and Peano.
   After an opening discussion (pages 1-84) in which the purposes, chief ideas and the material of the book are made clear, there is a division of the first part under two headings. The business of the pages under the first heading, 'Mathematical Logic', is with the theory of statement connections (pages 90-126), the theory of statements with variables (pages 127-186), and the theory of classes and relations as an algebra (pages 187- 326). The business of the pages under the second heading, ' Prolegomena to Cardinal Arithmetic ', is with ideas needed for giving a definition of 'cardinal number' and for building an arithmetic of cardinal numbers with the bricks of logic.
   The later, second and third parts go into the details of the arithmetic of cardinal and ordinal numbers, these arthmetics being based completely on (what the writers take to be only) logic.

11.2 Relation between Russell's ' Principles ' and 'Principia Mathematica'   The writing of Principia Mathematica was to take further the teaching of Russell's Principles of Mathematics. In the Principles Russell had put forward the opinion that all mathematics is -- may be got as -- the offspring of logic : all the ideas of mathematics may be given definitions using only ideas that .are a part of logic, and all the theorems of mathematics may be given demonstrations using only axioms and definitions that are a part of logic. The Principles is a long account and discussion of this view in the philosophy of mathematics. But it does not give a detailed development of logic with chains of deductions designed as the support of this view. This detailed development, as far as arithmetic, is offered in Principia Mathematica and that was the chief purpose of the writers' ten years' work on the book.

11.3 Russell's way of getting consistency in class theory.   In The Principles of Mathematics Russell made the attempt to overcome the doubts, whose seed was in the discoveries of Burali-Forti, himself and others, that the algebra of classes and the arithmetic of classes do not have the property of consistency, by ruling that what seem to be well-formed statements, such as 'x is an element of x', but which are the cause of trouble about consistency, are not well-formed ; somewhat like ' of is square a 4 root 2 ', they do not have good sense.
- 78 - A form of this same sort of ruling was taken up again in Principia Mathematica, and it has been much talked about as a way to get consistency in the theory of classes. Two sides of the ruling are firstly, that whatever is about all of a class is not able to be one of the class , and, secondly, that it is not possible for the values of a function to have parts which may be given definitions only in relation to the function itself. An effect of this is that the functions of which a given thing a may be an argument (8.5) are unable to be arguments of one another and that they have no things in common with the functions of which they may be arguments. In view of this one is forced to put functions into different levels; starting with a and the other things which may be arguments of the same functions as those of which a may be an argument, one comes to functions of which a is a possible argument, and then to functions of which such functions are possible arguments, and so on.
- 79 -

12   MATHEMATICAL LOGIC AFTER PRINCIPIA MATHEMATICS :
HILBERT'S METAMATHEMATICS


12.1. The growth of logic after 'Principia Mathematica'.   In the years shortly after Principia Mathematica came out one was working to make its axiom bases simpler and better. But there has been little agreement with the Frege-Russell-Whitehead opinion that mathematics is logic, though some who have been in agreement with it have been high authorities, for example Quine. Outside the algebra and arithmetic of classes as branches of everyday mathematics not formed as axiom systems, the chief business of the logic of mathematics in the last 50 to 60 years has been with the metamathematics of axiom systems of logic and mathematics and with the field of ideas given birth by metamathematics.

12.2 The structure of an axiom system.   The metamathematics of a branch of mathematics is generally done in connection with a more or less fully detailed axiom system of that branch. One became conscious of the conditions, and of the need, of a branch of mathematics being put in the form of a more or less fully detailed axiom system only by degrees, the present-day views being mostly the outcome of the writings of Pasch, Peano, Hilbert and others about axiom systems of simple arithmetic or of Euclid's and Lobachevsky's geometries, and of the writings of Frege, Couturat and the l920's Polish school of logic headed by Lesniewski, Lllkasiewicz and Tarski.
    In a fully detailed axiom system S, the letters and other signs of the language to be used in S are listed and rules are given saying which complexes of signs are well-formed statements in S ; there is a complete list of those well-formed statements which are to be used as axioms ; a complete list of used definitions is given ; there is an account of what is needed and what is enough for something to be a demonstration, and for something to be a theorem, in S ; there is a complete list of the rules of deduction in S, these rules making clear and limiting the moves that may be made in playing with the well-formed statements in S ; if the theorems of some other branch of logic or mathematics may be used in S, there is a statement saying which of such branches may be used as helps in S. Roughly, a well-formed statement is a theorem if and only if it is an axiom or a last step in a demonstration, a demonstration being a chain of well-formed statements every one of which is an axiom or rightly got by rules of deduction. If s is any well-formed statement in the language of S, and its opposite, having the sense ' s is false ', is -s, then S is said to have the property of consistency if and only if not the two of s and -s are theorems in S. S is said to be complete if and only if, for any well-formed statement s without free variables, at least one of s and -s is a theorem in S. If S does not have the property of consistency, it will be complete, because every s will become a theorem in S : any well-formed statement may be got as the outcome of two opposite theorems. In all other ways the properties of consistency and being complete are not at adl dependent on one another. One's normal purpose in building a fully detailed axiom system S is to get a system that has those two properties.

12.3. Post's, Hilbert and Ackermann's, and Goedel's demonstrations for ' Principia Mathematica ' systems.   Emil Post (1897-1954), while still a very young man, gave a demonstration in 1920 that the Principia Mathematica axiom system for the logic of statement connections has the property of consistency and that it is complete in the sense that every true law of that logic may be got as a theorem in the system. Being complete in this sense the system is, in addition, complete in the sense given in 12.2. Post's demonstration was based on a decision process that was in all important points the same as Peirce's (7.7). In addition, Post's 1920 paper had an outline account of his invention of an n-valued logic of statement connections, that is a logic in which not only the two values v and f but any number < i>n of values may be given to its statements. At the same time the invention of a 3-valued logic was made by Lukasiewicz; for example, if the three values are v (true), f (false) and u (uncertain), then the definitions of the values of Ns (= ' not-s ') and of s C t (= ' if s, then t ') are as in the tables :
	s Ns 	s t 	s C t
	v f 	v v	v 
	u u 	v u 	u
	f v 	v f 	f
		u v	v
		u u	v 
		u f 	u
		f v 	v
		f u 	v
		f f 	v

The n-valued and 3-valued logics of Post and Lukasiewicz are interesting in themselves and are important for certain parts of metamathematics.
    Later in the 1920's Hilbert and Ackermann gave a demonstration of the consistency of an axiom system of the logic of statements with variables, a system that was equal to the system in Principia Mathematica. And in 1930 Goedel gave a demonstration of the fact that the Principlia Mathematica system for this logic is complete . - 82 -

12.4 Brouwer's doubts and Hilbert's metamathematics.   The first demonstrations of the consistency of Euclid's geometry, and of Lobachevsky's geometry, were made in 1899, and in 1903, by the great German authority on mathematics David Hilbert (1862-1943). These demonstrations were made by the help of arithmetic. They have no value if arithmetic is without consistency. This condition of things where the consistency of one system S1 is dependent on the consistency of another system S2 is quite general. To give good reasons pointing to the consistency of a system, one has to go outside the system.
&bsp;   Again, no one had been able to get a consistency demonstration for the theory of classes. Because of the ever-present feeling that any branch of mathematics might not have consistency, a feeling made ever-present by the troubles in the theory of classes, the design of getting consistency demonstrations for arithmetic and class theory specially, and for other branches of mathematics, was slowly formed by Hilbert after 1899.
    Brouwer (1882-1966) had, from 1907, been responsible for the suggestion that something is seriously wrong with almost all reasoning in everyday mathematics, in so far as this makes use of classes that are not finite. In relation to such reasoning the ' law ' of logic named the tertium non datur (' there is no third value (like true, false)') is not true as a general law. The outcome of the belief in this law is that statements about numbers, points, and so on are somehow able to be true without being made, by man's thought, to be true by a building up of examples that make the statements true. But how are numbers, points, and so on to have existence separately from our minds ? Does it make sense to have the belief that they have existence till they are given existence by our building operations in mathematics ? And if it does not make sense, then it is not possible for statements about the ideas of mathematics, like numbers and points, to be true in themselves, as some strange sorts of facts, and separately from us.
    Hilbert made the decision to let only certain sorts of reasonings into his metamathematics, taking note of Brouwer's protests. It was, anyhow, unwise to make use, in a consistency demonstration, of any part of mathematics whose consistency itself was open to some doubt. One effect of this decision was that all Cantor's ordinals that are higher than some quite low level are not to be used in consistency demonstrations.
 
12.5 . Goedel's Theorem in the metamathematics of arithmetic.   In the 1920's Hilbert's chief purpose was to get a consistency demonstration of simple arithmetic -- of the arithmetic of the natural numbers. However, at the end of the 1920's -- in 1931 -- Goedel (1906-1978) made a most surprising and shocking discovery of a demonstration of a theorem in the metamathematics of simple arithmetic by which Hilbert's hopes were smashed. Goedel's Theorem itself did not come as a great surprise ; there had been earlier suggestions, for example in 1927 by von Neumann in an important paper on metamathematics demonstrations, that something like it was probably true. But experts were surprised that, with the current knowledge, a demonstration of it was able to be made.
    Goedel's Theorem is to the effect that no axiom system of simple arithmetic is complete if it has the property of consistency ; so whatever axiom system one has of simple arithmetic, with consistency, there is at least one true relation between natural numbers which may not be got as a theorem in it. - 84 -

12.6 The theory of recursive functions.   From the 1930's till now most work in metamathematics has been about arithmetic, being guided by the designs of Hilbert and by ideas used by Goedel, and by a number of different forms of these designs and ideas.
    Specially important has been the idea of that sort of function which is named recursive. Simple functions of this sort have been used for a long time in mathematics ; but there was no deep theory of this class of functions till after Goedel's Theorem whose demonstration made great use of recursive functions. A function f is said to be recursive if and only if, for any argument a of the class X of definition of f, X being made from the class N° of 0 and the natural numbers, the value b of f for a (8.5) may be fully worked out from a system of equations, the working out being done with the help of other values of f for other arguments and with the help of other functions whose definitions are given by the same system of equations. A very simple example of a recursive function is the power function whose definition is given by the equations (I) a°=1 , (I I) an+1"= a × an where n is an element of N°. The values of this function for any argument a may be worked out in this way : (1) a1=a × a° by (I I) , =a × 1 by (I), =a ; (2) a2= a × a1=a × a, by (1) ; and so on.
    The theory of recursive functions has deep connections with the theory of those functions all of whose values may be worked out by machines. Goedel, Skolem, Church, Kleene and Turing have been chiefly responsible for the development of these new theories. Accounts of them and of the other branches of Mathematical Logic that have been named in our discussion will be given in the other books coming out as Monographs in Modern Logic.
-85-

FURTHER READING

The purpose of this list is to give the names of some of the chief books on the history of Mathematical Logic. It is not to give the names of the writings of the makers of Mathematical Logic ; for these, see Church (3).
- 86 -

INDEX

Abel, N., 25-6 					Demonstration, 58, 74-5, 81
Ackermann, W., 82				De Morgan, A., 11, 34, 47-8 ,
Algebra, 3, 9, 11, 20, 23~8, 35, 504		Descartes, R., 9, 18-19, 21, 74
37, 40-4, 47, 49-50, 58				
Algebra of logic, 25, 37-59 			Diodorus, 13
Argument, 3, 5 					Duality, 57
Argument of function, 66 			Duhamel, G., 75
Aristotle, 3-12
Arithmetic, 18-20, 60-3, 68- 70, 83-5 		Element, 36, 75
						Enumerable, 67
Axiom, 32 					Equation, 23, 35, 46, 48
Axiom system, 10, 24, 28, 80			Euclid,10, 24, 28-30,32
						Excluded middle, 83
Bacon, F., 16 					Existence, 83
Basic English, 1-3
Bochenski, I. M., 86 				Fermat, P., 9
Boole, G., 9, 25, 34--44, 46-51, 57, 58, 77 	Finite, 67
						Free variable, 55
Brouwer, L E. ]., 83-4 				Frege, G., 22, 60-5, 67, 70,
Browning, R., 45 73-7, 79
Burali-Forti, C., 69, 71, 78 			Function, 63-6, 79, 85
Callimachus, 12 				Galois, E., 26
Cantor, G., 66-70, 77, 84 			General statements, 5, 34-5
Cardinal number, 62-3, 66- 70, 77-8		Geometry, 3, 9, 10, 20, 28-30, 80, 82-3
Carlyle, T., 45					Goedel, K., 82, 84-5
Cayley, A., 27 					Group theory, 26
Church, A., 85-6 				Hamilton, W., 34, 46
Class, 36-52, 62-3, 67-72, 75, 			Hilbert, D., 32, 80, 82-4 78-9 
Class of definition, 65				Huntington, E. V., 58 
Clifford, W. K., 45 				
Complete, 81 50, 73-4				Implication, 4, S, 10, 12-13
Computable function, 85 			Independence, 32, 58-9
Consistency, 31, 71-2, 78-9,			Interpretation, 38, 81-4
Czuber, E., 64-5 				Jevons, W. S., 44-9, 57
Dalgarno, G., 15-16				Jorgensen, J., 86
Darwin, C., 45 					Kant, I., 29
Decision process, 53 				Kleene, S. C., 85-6
Deduction, 3, 24, 31 				Kneale, M., 86
Definition, 52 					Kneale, W. C., 86
- 87 -

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